Springer Monographs in Mathematics

Kuranishi Structures and Virtual Fundamental Chains

Authors: Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.

Free Preview
  • Is the first book to provide a foundation of techniques of virtual fundamental chain by the originators of the theory 
  • Makes a timely appearance with details of the theory much needed for further development of the field 
  • Contains detailed explanations on orbifolds, collected in a single volume 
see more benefits

Buy this book

eBook $109.00
price for USA in USD
  • ISBN 978-981-15-5562-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • Immediate eBook download after purchase and usable on all devices
  • Bulk discounts available
Hardcover $139.99
price for USA in USD
About this book

The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures.
Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved.
Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures. 

Table of contents (29 chapters)

Table of contents (29 chapters)

Buy this book

eBook $109.00
price for USA in USD
  • ISBN 978-981-15-5562-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • Immediate eBook download after purchase and usable on all devices
  • Bulk discounts available
Hardcover $139.99
price for USA in USD
Loading...

Recommended for you

Loading...

Bibliographic Information

Bibliographic Information
Book Title
Kuranishi Structures and Virtual Fundamental Chains
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2020
Publisher
Springer Singapore
Copyright Holder
Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-15-5562-6
DOI
10.1007/978-981-15-5562-6
Hardcover ISBN
978-981-15-5561-9
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XV, 638
Number of Illustrations
115 b/w illustrations, 34 illustrations in colour
Topics